#499911

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(6,12)(7,16)(8,10)(9,21)(13,27)(14,31)(15,19)(17,34)(18,33)(20,23)(22,24)(25,28)(29,36)(30,32);(1,2)(3,4)(5,6)(7,13)(8,19)(9,14)(10,12)(11,15)(16,28)(17,29)(18,30)(20,35)(21,26)(22,32)(23,31)(24,25)(27,34)(33,36);(2,4)(5,7,11,18)(6,13,15,30)(8,17,10,24)(9,22,23,34)(12,25,19,29)(14,32,31,27)(16,20,33,21)(26,28,35,36);(5,6)(7,13)(8,19)(9,14)(10,12)(11,15)(16,28)(17,29)(18,30)(20,35)(21,26)(22,32)(23,31)(24,25)(27,34)(33,36)

Polycyclic presentation:

2^9|c^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,e^b=ei,f^c=fh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407107);